On variable exponent Lebesgue spaces of entire analytic functions

Motos, Joaquín; Planells, María Jesús; Talavera, C.F.

doi:10.1016/j.jmaa.2011.09.069

Cited by 8 publications

(9 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We shall denote by the collection of all ∈ such that supp̂⊂ ; obviously ⊂ (⋅) . The spaces (⋅) have been introduced and studied in [4]. …”

Section: Notationmentioning

confidence: 99%

“…(Each step (⋅) ∩ E ( ) is a quasi-Banach space since it is isomorphic to − (⋅) via the Fourier transform and this space is a quasi-Banach space by [4,Theorem 3.5]. On the other hand, the bilinear mapping ×( ∩E ( )) → ∩E ( ) : ( , ) → is continuous (see [5])).…”

Section: Notationmentioning

confidence: 99%

“…For a thorough discussion of these spaces and their history, see [1,2]. Our paper lies in this field of variable exponent function spaces and is a continuation of [3] (see also [4,5]). In [5] the (nonweighted) variable exponent Hörmander spaces (⋅) , (⋅) (Ω), and loc (⋅) (Ω) were introduced (recall that the classical Hörmander spaces , , , (Ω), and loc , (Ω) play a crucial role in the theory of linear partial differential operators (see, e.g., [6][7][8][9][10])) and there, extending a Hörmander result [6, Chapter XV] to our context, the dual of (⋅) (Ω) (when 1 < − ≤ + < ∞) was calculated (as a consequence some results on sequence space representation of variable exponent Hörmander spaces were obtained).…”

Section: Introduction and Notationmentioning

confidence: 99%

“…The techniques used in the article (also in [3]) are based on the inequalities of the extrapolation theorems obtained by the authors in [4] and on the properties of the Banach envelopes of the 0 -Banach local spaces of loc (⋅) (Ω). Finally, three questions on duality and on sequence space representation of variable exponent Hörmander spaces are proposed.…”

Section: Introduction and Notationmentioning

confidence: 99%

See 3 more Smart Citations

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

Motos

Planells

Talavera

2016

Abstract and Applied Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…We shall denote by the collection of all ∈ such that supp̂⊂ ; obviously ⊂ (⋅) . The spaces (⋅) have been introduced and studied in [4]. …”

Section: Notationmentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

Section: Introduction and Notationmentioning

confidence: 99%

Section: Introduction and Notationmentioning

confidence: 99%

See 2 more Smart Citations

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

Motos

Planells

Talavera

2016

Abstract and Applied Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then we introduce and study an important locally convex topology on B c p(•) (Ω ) (considering the Banach envelopes of those steps) and we show that the space B loc ∞ (Ω ) is isomorphic to B c p(•) (Ω ) (this is the main result of the paper). The estimates obtained in [16,Theorem 3.5] play an essential role in the proof of this isomorphism. As a consequence of this result, we obtain a sequence space representation of the dual B c p(•) (Ω ) improving a result of [17] (the corresponding results for p(•) ≡ p, 0 < p < 1, are also new).…”

Section: Introductionmentioning

confidence: 95%

Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ 0 < p - ≤ p + ≤ 1 ) and some applications

Motos

Planells

Talavera

2014

RACSAM

Self Cite

View full text Add to dashboard Cite

In this paper we characterize the dual B c p(•) (Ω) of the variable exponent Hörmander space B c p(•) (Ω) when the exponent p(•) satisfies the conditions 0 < p − ≤ p + ≤ 1, the Hardy-Littlewood maximal operator M is bounded on L p(•)/p 0 for some 0 < p 0 < p − and Ω is an open set in R n. It is shown that the dual B c p(•) (Ω) is isomorphic to the Hörmander space B loc ∞ (Ω) (this is the p + ≤ 1 counterpart of the isomorphism B c p(•) (Ω) B loc p (•) (Ω), 1 < p − ≤ p + < ∞, recently proved by the authors) and hence the representation theorem B c p(•) (Ω) l N ∞ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of B c p(•) (Ω) and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for p(•) ≡ p, 0 < p < 1, are also given (e.g. B c p (Ω) does not contain any infinite-dimensional q-Banach subspace with p < q ≤ 1 or the quasi-Banach space B p ∩ E (Q) contains a copy of l p when Q is a cube in R n). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hörmander spaces is proposed. Keywords Variable exponent • Hardy-Littlewood maximal operator • Banach envelope • L p(•)-spaces of entire analytic functions, Hörmander spaces Mathematics Subject Classification (2000) 46F05 (46E50) • 46A16 • 42B25

show abstract